Toward the proof of differential transcendence, we often need to prove that the difference Galois group is big, making important to be able to compute the latter. However, the computation of difference Galois groups is in general a difficult task. In the literature one can find many algorithms for the computation of Galois groups. The most developed area is the one of differential equations: many algorithms are given to compute differential Galois groups (which are algebraic groups attached to a linear differential system), see for example [1]}, [2]}, [3]}, [4]}, [5]}.
There is a general algorithm for the computation of the Galois groups of linear differential equations of an arbitrary order but it is not effective (see for example the algorithm of Hrushovski in [6]} and some improvements [7]}, [8]}). In contrast, the case of difference equations is less developed.
Algorithms have been given for the computation of difference equations of order 1 and 2 in these four famous cases: the shift (see [9]}, [10]}), the \(q\) -difference (see [11]}), the Mahler (see [12]}) and the elliptic cases (see [13]}). For the shift case, an analogue of Hrushovski's algorithm has been developed in [14]}. It computes the Galois groups for equations of arbitrary order whose coefficients are rational functions but is still inefficient. However, in general, for the Galois groups of difference equations of order 3 or more, only few things are known. The aim of this article is to explain how one can compute Galois groups of difference equations of order 3 under mild assumptions on the difference field to which the coefficients of the difference equations belong. In particular, it takes into account the four previous difference equations. We highlight that we are mainly concerned by the theoretical aspects of the algorithm and no study of complexity will be made. Throughout this paper, we explain what we need to be able to do for using it in practice. In fact, the key point to apply it concretely is to be able to find rational (resp. hyperexponential) solutions to certain difference equations.
